When the Odds Decide: Sensitivity and Risk

Lesson 2 of 2: Break-Even and the Limits of Expected Value

In this lesson:

• Recompute the comparison under different odds
• Find the break-even accident probability
• See when risk overrides expected cost

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Recompute a strategy comparison under different reasonable probabilities
2. Find the break-even probability where the recommendation changes
3. Explain when risk justifies a higher-expected-cost choice

Our Answer Rested on an Assumption

We recommended Policy B — for a driver with accident odds .

But what about a driver who crashes far more often?

Try Various, But Reasonable, Chances

The standard asks you to try different reasonable odds — not just one.

The better policy is a function of the accident probability, not a fixed verdict.

Recompute for a Higher-Risk Driver

At accident probability : , . B still wins, but barely.

Write Each Cost as a Function of p

Let be the accident probability.

Set the Two Costs Equal

The break-even is where the policies cost the same:

Solve for the Break-Even Probability

Report a Condition, Not a Verdict

The complete answer:

Choose Policy B if your accident probability is below about 0.53; otherwise choose Policy A.

Your Turn: A Warranty Break-Even

A phone repair costs $600. A warranty costs $90 up front and covers it.

Find the repair probability where the two strategies tie.

Set warranty cost = expected self-insure cost.

But Expected Cost Isn't the Whole Story

We've been minimizing expected cost.

Yet people rationally buy insurance that costs more on average. Why?

Insurance Costs More — and Is Still Rational

Buying insurance has a higher expected cost than self-insuring.

You pay a small certain amount to cap a rare, ruinous loss.

Same Average, Very Different Worst Case

Two strategies can share a mean but differ wildly in their worst outcome.

Risk Can Override the Average

• Expected cost is the primary decision criterion
• But the size of possible losses — risk — also matters
• A risk-averse person may accept a higher average to cap the worst case

Your Turn: When Does Risk Win?

A business can take a steady plan or a risky plan with the same expected profit, but the risky one has a small chance of bankruptcy.

Which should a small, cash-poor company choose — and why?

What You Built In This Lesson

✓ Recompute under new odds; find the break-even probability

✓ Report a condition, not a single verdict

✓ Expected cost is primary — but risk can override it

Coming Up Next: Full Decision Analysis

The next standard adds conditional probability and asymmetric error costs to the frame.

You'll analyze medical tests, product testing, and a coach pulling the goalie.